Question 1. How can one distinguish
the sphere from a plane (Flatland) based solely on experiences on the surface?
Question 2. How can one distinguish
the sphere from a torus
based solely on experiences on the surface?
Show video on PT. Starting with "pythagoras"
Outline:
Area of triangles = 1/2 bh
Area of parallelogram= bh
Scaling: a linear scale change of r gives area change of factor r^{2}.
3 questions: running, moat, wind power...
Proof of the PT: Similar right triangles: c= a^{2 }/c + b^{2 }/c .
applications and other proofs.
Prop. 47 of Euclid.
Dissection Proof.
Prop 31 Book VI Similar shapes.
Simple proof of PT using similar triangles of the triangle.
Use in 3 dimensional space.